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International Journal of Engineering Research-Online
A Peer Reviewed International Journal Email:[email protected] http://www.ijoer.in
Vol.4., Issue.3., 2016 (May-June)
226 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
INTRODUCTION
Cross beams are provided mainly to stiffen
the girders and to reduce torsion in the exterior
girders. These are essential over the supports to
prevent lateral spread of the girders at the bearings.
Another function of the cross beams is to equalize
the deflections of the girders carrying heavy loading
with those of the girders with less loading. The
thickness of the cross beam should not be less than
the minimum thickness of the webs of the
longitudinal girders. The depth of the end cross
girders should be such as to permit access for
inspection of bearing and to facilitate positioning of
jacks for lifting of superstructure for replacement of
bearings.
RESEARCH ARTICLE ISSN: 2321-7758
ALTERNATIVE FORMS OF T-BEAM BRIDGE SUPERSTRUCTURE FOR FOUR LANE
BRIDGES – A STUDY BY GRILLAGE ANALOGY
Prof.M.G.KALYANSHETTI1, K.S.BHOSALE2
1Department of Civil Engineering
Walchand Institute of Technology, Solapur, Maharashtra State, India 2 Asst.Professor, Department of Civil Engineering
NBN Sinhgad College of Engg, Solapur, Maharashtra State, India
ABSTRACT
This paper concern with effectiveness of cross beams on T-Beam Bridge. Cross
beams are provided mainly to stiffen the girders and to reduce torsion in the
exterior girders. These are essential over the supports to prevent lateral spread of
the girders at the bearings. Another function of the cross beams is to equalize the
deflections of the girders carrying heavy loading with those of the girders with less
loading. The thickness of the cross beam should not be less than the minimum
thickness of the webs of the longitudinal girders. The depth of the end cross girders
should be such as to permit access for inspection of bearing and to facilitate
positioning of jacks for lifting of superstructure for replacement of bearings.
Prior to 1956, T-beam bridges had been built without any cross beams or
diaphragms, necessitating heavy ribs for the longitudinal beams. In some cases, only
two cross beams at the end have been used. The provision of cross beams facilitates
adoption of thinner ribs with bulb shape at bottom for the main beams. The current
Indian practice is to use one cross beam at each support and to provide one to three
intermediate cross beams. Diaphragms have been used instead of cross beams in
some cases in the past.
Hence separate study is required for various spans of two lane and four lane bridges
with different IRC standard live loads to understand effectiveness of cross beams to
stiffen the girders and to reduce torsion in the exterior girders which will lead to
effective selection of cross beams.
Key words: Indian Road Congress, Grillage Analogy, Four lane bridges, Cross girders,
RC Live Loads, T-Beam Bridge ©KY Publications
International Journal of Engineering Research-Online
A Peer Reviewed International Journal Email:[email protected] http://www.ijoer.in
Vol.4., Issue.3., 2016 (May-June)
227 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
Prior to 1956, T-beam bridges had been built
without any cross beams or diaphragms,
necessitating heavy ribs for the longitudinal beams.
In some cases, only two cross beams at the end have
been used. The provision of cross beams facilitates
adoption of thinner ribs with bulb shape at bottom
for the main beams. The current Indian practice is to
use one cross beam at each support and to provide
one to three intermediate cross beams. Diaphragms
have been used instead of cross beams in some
cases in the past.
PROCEDURE
(I) It is easy for an engineer to visualize and prepare
the data for a grillage. Grillage Analogy is based on
stiffness matrix approach and was made amenable
to computer programming by Lightfoot and Sawko
in 1959. West conducted experiments on the use of
grillage analogy in 1973. He made suggestions
towards geometrical layout of grillage beams to
simulate a variety of concrete slab and pseudo-slab
bridge decks, with illustrations. Gibb developed a
general computer program for grillage analysis of
bridge decks using direct stiffness approach that
takes into account the shear deformation also in
1972. Martin in 1996, then followed by Sawko
derived stiffness matrix for curved beams and
proclaimed a computer program for a grillage for
the analysis of decks, curved in plan in 1967. The
grillage analogy has also been used by Jaeger and
Bakht for a variety of bridges in 1982.
(II) Grillage Analogy
There are essentially five steps to be followed for
obtaining design responses:
I. Idealization of physical deck into equivalent
grillage
II. Evaluation of equivalent elastic inertias of
members of grillage
III. Application and transfer of loads to various
nodes of grillage
IV. Determination of force responses and
design envelopes and
V. Interpretation of results.
The method consists of ‘converting’ the bridge deck
structure into a network of rigidly connected beams
at discrete nodes i.e. idealizing the bridge by an
equivalent grillage as shown in figure1. The
deformations at the two ends of a beam element
are related to the bending and torsional moments
through their bending and torsional stiffnesses.
These moments are written in terms of the
end-deformations employing slope-deflection and
torsional rotation-moment equations. The sheer
force in the beam is also related to the bending
moment at the two ends of the beam and can again
be written in terms of the end-deformations of the
beam. The shear and moment in all the beam
elements meeting at a node and fixed end reactions,
if any, at the node, are summed-up and three basic
statical equilibrium equations at each node namely
∑Fz = 0, ∑Mx = 0 and ∑My = 0 are satisfied.
In general a grid having ‘n’ nodes will have
‘3n’ nodal deformations and ‘3n’ equilibrium
equations relating to these. Back substitution in the
slope-deflection and torsional rotation-moment
equations will give the bending and torsional
moments at the two ends of each beam element.
Shear forces are computed from bending moments
and external loads.
EXPERIMENT AND RESULT: A model of T-beam
bridge deck is prepared on STAADPro software and
analyzed by Grillage Analogy. The detailed study is
carried out for Four Lane Bridges of spans 15m,
20m, 25m, 30m, 35m and 40m with live loads as IRC
ClassA and IRC 70R loading. For each span, three
systems are considered as follows:
1) Three cross girders system (3 CG)
2) Five cross girders system (5 CG)
3) Seven cross girders system (7 CG)
To illustrate the grillage analogy in bridge deck
analysis, detailed calculations are shown for T-beam
bridge for following data.
Data:
1) Clear Roadway = 15.2m (Four Lane Bridge)
2) Span of T – Beam = 20m
3) No. of Longitudinal girders = 5
4) c/c of Longitudinal girders = 2.5m
5) Thickness of deck slab = 215mm
6) Thickness of wearing coat = 75mm
7) Web thickness of main & cross girders = 250mm
8) Width & depth of Long girder = 300mm, 1.6m
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228 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
9) Width & depth of cross girders = 250mm, 1.28m
10) Live Load= 4 trains of IRC class A loading, 2
trains of IRC class 70R loading.
Calculations for sectional properties are done and
shown in table-1.
Table - 1 Properties of grillage lines of 20m span four lane bridge with 3cross girders.
After analyzing the grillage models, results are tabulated as shown in table - 2 & 3.
Table 2: Maximum Dead Load and Live Load BM, Deflection and Torsion in each longitudinal girder with IRC
class A-4 trains loading for 20 m span four lane bridge with 3 cross girders.
Long. Girde
r
D L B M kNm
L L B M kNm
Total BM kNm
D L defl. mm
L L defl. mm
Total defl. mm
Defl / span
D L Torsion
kNm
L L Torsion
kNm
Total Torsion
kNm
1 2483.050 1217.543 3700.593 17.877 8.100 25.977 1/769 56.28 41.83 98.11
2 2779.514 1507.297 4286.811 15.909 8.219 24.128 1/828 31.66 26.36 58.02
3 2620.859 1511.867 4132.726 15.198 8.234 23.432 1/853 0.00 19.27 19.27
4 2779.514 1507.297 4286.811 15.909 8.219 24.128 1/828 31.66 27.47 59.13
5 2483.050 1217.543 3700.593 17.877 8.100 25.977 1/769 56.28 39.58 95.86
Grillage Line Section Area(m²) Ix (m4) Iz (m4) Dire-ction
1,11
0.7300 0.0097 0.0335 Longi.
2,10
1.0688 0.2682 0.0497 Longi.
4,6,8
1.3375 0.3269 0.0544 Longi.
12,24
0.3652 0.0551 0.0058 Lateral
18
0.4103 0.0640 0.0065 Lateral
13-17, 19-23
0.4012 0.0016 0.0057 Lateral
3,5,7,9 (Dummy) 0 10−7 10−7 Longi.
International Journal of Engineering Research-Online
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229 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
Table 3: Maximum Dead Load and Live Load BM, Deflection and Torsion in each longitudinal girder with IRC
class 70R-2 trains loading for 20 m span four lane bridge with 3 cross girders.
Long.
Girder
D L B M
kNm
L L B M
kNm
Total BM
kNm
D L defl.
mm
L L defl.
mm
Total defl.
mm
Defl /
span
D L
Torsion
kNm
L L
Torsion
kNm
Total
Torsion
kNm
1 2483.050 1079.123 3562.173 17.877 6.380 24.257 1/824 56.28 120.34 176.62
2 2779.514 1784.846 4564.360 15.909 10.137 26.046 1/767 31.66 132.96 164.62
3 2620.859 2085.015 4705.874 15.198 11.761 26.959 1/741 0 150.86 150.86
4 2779.514 1784.846 4564.360 15.909 10.137 26.046 1/767 31.66 95.56 127.22
5 2483.050 1079.123 3562.173 17.877 6.380 24.257 1/824 56.28 54.10 110.38
Figure1. Idealization of bridge deck into equivalent grillage
A typical grillage layout in STAAD is shown in figure3.
Typical bending moment variation, deflection
variation and torsion variation is shown in figure 4, 5
and 6 respectively.
A parametric study is carried out for spans of 15m to
40m for IRC class A loading. Variation in the bending
moment, deflection and torsion is studied and are
presented in figure 7, 8 and 9.
The same study is carried out for IRC class 70R
loading and results are presented in figure10, 11 and
12 for which above all three systems of
superstructure are considered. i.e. 3 CG, 5 CG and 7
CG.
Figure2. Typical Grillage Layout with Loading
Figure3. Grillage Layout for 20m span four lane
bridge with 3 cross girders
Figure4. Typical Maximum Bending Moment
Diagram
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230 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
Figure5. Typical Maximum Deflection Diagram
Figure6. Typical Maximum Torsion Diagram
Figure7.Variation in maximum B M in Longitudinal girders w.r.t. span for four lane bridge with IRC class A-4
trains of loading.
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Figure8. Variation in maximum Deflection in longitudinal girders w.r.t. span for four lane bridge with IRC class
A-4 trains of loading.
Figure9. Variation in maximum Torsion w.r.t. span for four lane bridge with IRC class A-4 trains of loading.
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232 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
Figure10. Variation in maximum B M in longitudinal girders w.r.t. span of four lane bridge with IRC 70 R two
trains of loading.
Figure11. Variation in maximum Deflection in longitudinal girders w.r.t. span of four lane bridge with IRC 70 R
two trains of loading.
International Journal of Engineering Research-Online
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233 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
Figure12. Variation in maximum Torsion w.r.t. span of four lane bridge with IRC 70 R two trains of loading.
CONCLUSION
1 Bending Moment in longitudinal girders –
With the increase in number of cross
girders, self weight of bridge increases and
hence maximum bending moment
increases. Rate of increase in BM is mild
upto 25m, beyond that, rate increases.
Thus for higher span i.e. greater than 25m,
BM in longitudinal girder increases with
higher rate.
The same trend is observed for both IRC
classA and IRC class70R loading with almost
same bending moment.
2 Deflection in longitudinal girders –
Cross beams equalizes the deflections of
the girders carrying heavy loading with
those of the girders with less loading.
Hence in all girders deflection is almost
same. With the increase in number of cross
girders, deflection increases.
Rate of increase in deflection is more upto
25 to 30m. Beyond this, rate decreases.
The same trend is observed for both live
load. This is because the length of these
vehicles (IRC classA, IRC class70R) are in the
range of 20 to 25m. Beyond the span of
25m, contribution of live load to deflection
reduces. Hence beyond 25 to 30m, rate of
increase in BM reduces.
3 Torsion in longitudinal girders –
For four lane bridges, 5 longitudinal girders
are used. For increased number of
longitudinal girders, cross girders play very
vital role. In our case, it is observed that
from 3 CG to 7 CG, torsion in the exterior
girders tremendously reduced by almost 80
to 100% for higher spans (25 to 40m) and
almost 50 to 70% for spans upto 25m.
Thus it is concluded that for higher number
of longitudinal girders, effectiveness of
cross girders is increased manifolds.
REFERENCES
[1]. Johnson Victor, Essentials of Bridge
Engineering, Oxford and IBH Publishing
company Pvt. Ltd, New Delhi.
[2]. Surana C.S., Agrawal R. (1998). “Grillage
Analogy in Bridge Deck Analysis”, Narosa
Publishing House.
[3]. Indian Standard Code of Practice for Plain
and Reinforced Concrete, IS: 456 – 2000,
Indian Standard Institution, New Delhi.
[4]. V.K. Raina (1994), Concrete Bridge Practice-
Analysis, Design and Economics, Tata
McGraw –Hill Publishing Company Ltd.,
New- Delhi.
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A Peer Reviewed International Journal Email:[email protected] http://www.ijoer.in
Vol.4., Issue.3., 2016 (May-June)
234 Prof.M.G.KALYANSHETTI, K.S.BHOSALE
[5]. Hambly E.C. (1991). “Bridge Deck
Behaviour”, E & FN Spon.
[6]. IRC:5-1998-Standard Specifications and
Code of Practice for Road Bridges, Section I
– General Features of Design (Seventh
Revision)
[7]. IRC:6-2000-Standard Specifications and
Code of Practice for Road Bridges, Section II
– Loads and Stresses (Fourth Revision)
[8]. IRC:21-2000 Standard Specifications and
Code of Practice for Road Bridges, Section
III – Cement Concrete (Plain and
Reinforced) (Third Revision)